Let us define S=(∪i∈IAi)′
and T=∩i∈IAi′. To establish the equalityS=T, we shall
use a standard argument for proving equalities in set theory. Namely,
we show that S⊂T and T⊂S.
For the first claim, suppose x is an
element in S.
Then x∉∪i∈IAi, so x∉Ai for any i∈I.
Hence x∈Ai′ for all i∈I, and x∈∩i∈IAi′=T.
Conversely, suppose x is an
element in T=∩i∈IAi′. Then x∈Ai′ for all i∈I.
Hence x∉Ai for any i∈I, so x∉∪i∈IAi,
and x∈S.